Theorem int0 4919

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4453	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3446	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4911	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 231	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 264	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2734	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  ∅c0 4287  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-dif 3906  df-nul 4288  df-int 4905

This theorem is referenced by:  unissint  4929  uniintsn  4942  rint0  4945  intex  5291  intnex  5292  oev2  8460  fiint  9239  elfi2  9329  fi0  9335  cardmin2  9923  00lsp  20944  cmpfi  23364  ptbasfi  23537  fbssint  23794  fclscmp  23986  zarcmplem  34058  rankeq1o  36384  bj-0int  37345  heibor1lem  38049  ipoglb0  49342  mreclat  49345